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Maxwell-Boltzmann Distribution Calculator

Enter molar mass, temperature, and speed to evaluate the Maxwell-Boltzmann distribution.

How it is calculated

The tool first converts the molar mass M (g/mol) to the mass of one molecule, m = (M/1000) ÷ NA in kg, using Boltzmann's constant k = 1.380649×10-23 J/K. The normalised speed density is f(v) = 4π × (m / 2πkT)3/2 × v² × e−mv²/2kT, with T in kelvin and v in m/s; its unit is s/m, and it is a density per unit speed, not a probability. The three characteristic speeds are vp = √(2kT/m), vmean = √(8kT/πm) and vrms = √(3kT/m). Molar mass and temperature must be above zero and speed zero or more. The result assumes an ideal gas in thermal equilibrium.

Worked example

Nitrogen (28 g/mol) at 300 K, evaluated at v = 500 m/s. The density is 1.8441e-3 s/m, i.e. 0.0018441 per m/s. Multiplied by a narrow band of 10 m/s around 500 m/s, that is about 0.018, so roughly 1.8% of the molecules move at 500 to 510 m/s. The characteristic speeds are vp = 422.10 m/s, vmean = 476.29 m/s and vrms = 516.96 m/s, in increasing order.

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Common mistakes

  • Entering the temperature in degrees Celsius. The field is absolute temperature in kelvin; typing 27 for a 27 °C room gives results for a 27 K gas, with speeds of about 127 m/s instead of 422 m/s.
  • Entering the molar mass in kg/mol instead of g/mol. Nitrogen is 28, not 0.028.
  • Reading the density as a probability. It is per unit speed (s/m), so multiply it by a speed interval to get a fraction of molecules.
  • Confusing the three speeds. The most probable, mean and rms speeds differ, and the rms value is the one tied to the average kinetic energy.
  • Using the formula for a gas that is not ideal or not in equilibrium, such as a very dense gas or a beam from a nozzle.

About this calculator

The Maxwell-Boltzmann distribution describes how molecular speeds are spread across a gas at thermal equilibrium: f(v) = 4π (m/2πkT)^(3/2) v² e^(−mv²/2kT), where m is the molecule's mass, T the absolute temperature, and k the Boltzmann constant. This f(v) is a normalised probability density (it integrates to 1 over all speeds), so it gives the fraction of molecules per unit speed rather than a molecule count. From this single distribution three characteristic speeds fall out algebraically — the most probable speed v_p = √(2kT/m), the mean speed v_mean = √(8kT/πm), and the root-mean-square speed v_rms = √(3kT/m), each slightly larger than the last.

This calculator evaluates the probability density at a chosen speed and returns all three characteristic speeds for a given molar mass and temperature. Physics and physical chemistry students use it to check kinetic theory homework, while engineers modeling gas diffusion, effusion or reaction rates and atmospheric scientists studying how light gases like hydrogen escape a planet's atmosphere rely on the same speed distribution.

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